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Chapter 6: Problem 21

Maximum field in a cyclotron ** For some purposes it is useful to accelerate negative hydrogen ions in a cyclotron. A negative hydrogen ion, \(\mathrm{H}^{-}\), is a hydrogen atom to which an extra electron has become attached. The attachment is fairly weak; an electric field of only \(4.5 \cdot 10^{8} \mathrm{~V} / \mathrm{m}\) in the frame of the ion (a rather small field by atomic standards) will pull an electron loose, leaving a hydrogen atom. If we want to accelerate \(\mathrm{H}^{-}\)ions up to a kinetic energy of \(1 \mathrm{GeV}\left(10^{9} \mathrm{eV}\right)\), what is the highest magnetic field we dare use to keep them on a circular orbit up to final energy? (To find \(\gamma\) for this problem you only need the rest energy of the \(\mathrm{H}^{-}\)ion, which is of course practically the same as that of the proton, approximately \(1 \mathrm{GeV}\).)

Short Answer

Expert verified
The maximum magnetic field that could be used to keep the negative hydrogen ions on a circular path up to the final energy without exceeding the limit where they would be stripped of an extra electron is given by \(B_{max} = 2mc / qr\).

Step by step solution

01

Understand the cyclotron concept

The cyclotron is a type of particle accelerator which is used to accelerate charged particles to high energies. It does this by using a constant magnetic field to keep the particles moving along a circular path, while an oscillating electric field is used to increase their kinetic energy. The kinetic energy of an accelerated particle in cyclotron is given by \(E= \gamma mc^2\), with \(E\) as the kinetic energy, \(m\) the particle's mass, \(c\) the speed of light, and \(\gamma\) the Lorentz factor
02

Determine the Lorentz factor

The Lorentz factor (\(\gamma\)) is a function of the velocity of the particle and is given by the equation \(\gamma = 1/ \sqrt{1-(v/c)^2}\) where \(v\) is the velocity of the particle and \(c\) is the speed of light. In this problem, it was provided that the rest energy of the ion is approximately \(1 GeV\), which is the same as the kinetic energy of the ion. Therefore, \(\gamma\) can be considered approximately equal to 2.
03

Calculate the magnetic field

In a cyclotron, the centripetal force providing the circular motion of the ions is supplied by the magnetic force, given by \(F = qvB\), where \(q\) is the charge of the particle, \(v\) its speed, and \(B\) the magnetic field. Equating this with the centripetal force \(F = m\gamma v^2 / r\), we can solve for \(B\), obtaining \(B = \gamma mc / qr\). Now, by assuming maximum speed saturation \(v \approx c\), the magnetic field can be evaluated at this extreme condition to avoid electric field limit, obtaining a maximum magnetic field \(B_{max} = 2mc / qr\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Particle Accelerator
A particle accelerator is a device used to increase the speed and energy of particles such as electrons or ions. These devices are key in both research and practical applications, such as in physics experiments or medical treatments.
Cyclotrons are a specific type of particle accelerator that use a magnetic field to keep particles moving along a circular trajectory and an electric field to increase their velocity.
As particles are charged and move through the magnetic fields, they gain energy from the electric field, thus gaining speed.
  • Linear accelerators: particles move in a straight line.
  • Circular accelerators: particles move in a circular path, like cyclotrons.
A cyclotron's capability to increase a particle's kinetic energy makes it vital for detailed studies in particle physics.
Lorentz Factor
The Lorentz factor, denoted as \(\gamma\), is a crucial concept in special relativity and plays an essential role in understanding how time and energy behave at relativistic speeds.
It is defined by the equation \(\gamma = 1/ \sqrt{1-(v/c)^2}\), where \(v\) represents the velocity of the particle and \(c\) the speed of light.
As the speed of a particle approaches the speed of light, \(\gamma\) increases significantly, indicating relativistic effects become more pronounced. Here are some important points:
  • The Lorentz factor modifies the kinetic energy formula for particles moving at significant fractions of the speed of light.
  • In this problem, the rest energy is approximately similar to \(1 \, \text{GeV}\), suggesting that \(\gamma\) equals roughly 2 at the given conditions.
This adjustment is important for accurately calculating the required energy for particles in accelerators like cyclotrons.
Kinetic Energy
Kinetic energy in the context of a cyclotron is the energy gained by particles as they are accelerated. It is quantified as the amount of work needed to move the particle to a specific velocity.
The practical formula used in relativistic contexts combines mass, speed of light, and the Lorentz factor: \(E = \gamma mc^2\).
  • Where \(E\) stands for kinetic energy.
  • \(\gamma\) is the Lorentz factor.
  • \(m\) represents the mass of the particle.
In this exercise, the fact that the hydrogen ion \(H^-\) needs to reach a kinetic energy of \(1 \, \text{GeV}\) is the working condition to arrange the cyclotron's specifications suitably.
The requirement for such specific energy levels necessitates accurate calculations and adjustments in the cyclotron's magnetic field setup.
Magnetic Field
The magnetic field in a cyclotron is responsible for maintaining particles on a stable circular path while they are being accelerated.
It provides the necessary centripetal force to counteract their tendency to move in a straight line due to their inertia. The expression \(F = qvB\) describes the magnetic force, where \(q\) is the particle's electric charge, \(v\) its velocity, and \(B\) the strength of the magnetic field.
  • The essential balance accounts for the magnetic force in correspondence with the centripetal force.
  • For this system, \(F = m\gamma v^2 / r\) equates to \(qvB\), allowing the calculation of the optimum magnetic field.
  • With maximum speed \(v \approx c\), the field reaches its extreme, ensuring particles remain within the cyclotron's designated path.
This dynamic manipulation of the magnetic field is central to managing particle trajectories and speeds effectively in a cyclotron setting.

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Most popular questions from this chapter

Vector potential inside a wire ** A round wire of radius \(r_{0}\) carries a current \(I\) distributed uniformly over the cross section of the wire. Let the axis of the wire be the \(z\) axis, with \(\hat{z}\) the direction of the current. Show that a vector potential of the form \(\mathbf{A}=A_{0} \hat{\mathbf{z}}\left(x^{2}+y^{2}\right)\) will correctly give the magnetic field \(\mathbf{B}\) of this current at all points inside the wire. What is the value of the constant, \(A_{0}\) ?

Field at the center of a disk * A disk with radius \(R\) and surface charge density \(\sigma\) spins with angular frequency \(\omega\). What is the magnetic field at the center?

Helmholtz coils One way to produce a very uniform magnetic field is to use a very long solenoid and work only in the middle section of its interior. This is often inconvenient, wasteful of space and power. Can you suggest ways in which two short coils or current rings might be arranged to achieve good uniformity over a limited region? Hint: Consider two coaxial current rings of radius \(a\), separated axially by a distance \(b\). Investigate the uniformity of the field in the vicinity of the point on the axis midway between the two coils. Determine the magnitude of the coil separation \(b\) that for given coil radius \(a\) will make the field in this region as nearly uniform as possible.

A slab and a sheet \(* *\) A volume current density \(\mathbf{J}=J \hat{\mathbf{z}}\) exists in a slab between the infinite planes at \(x=-b\) and \(x=b\). (So the current is coming out of the page in Fig. 6.37.) Additionally, a surface current density \(\mathcal{J}=2 b J\) points in the \(-\hat{\mathbf{z}}\) direction on the plane at \(x=b\) (a) Find the magnetic field as a function of \(x\), both inside and outside the slab. (b) Verify that \(\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}\) inside the slab. (Don't worry about the boundaries.)

Motion in \(E\) and B fields \(* * *\) The task of Exercise \(6.29\) is to show that if a charged particle moves in the \(x y\) plane in the presence of a uniform magnetic field in the \(z\) direction, the path will be a circle. What does the path look like if we add on a uniform electric field in the \(y\) direction? Let the particle have mass \(m\) and charge \(q\). And let the magnitudes of the electric and magnetic fields be \(E\) and \(B\). Assume that the velocity is nonrelativistic, so that \(\gamma \approx 1\) (this assumption isn't necessary in Exercise 6.29, because \(v\) is constant there). Be careful, the answer is a bit counterintuitive.
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